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\title{Math 23a Problem Set 11}
\begin{document}
\maketitle

Let $V$ be a finite-dimensional vector space over $\bR$.We denote by
$C^\infty (V)$ the space of functions $f$ on  $V$ such 
that  $f$ has continuous derivatives  of all orders. In particular we
denote by $C^\infty (\bR)$ the space of functions $f$ on  $\bR$ such 
that  $f$ has continuous derivatives  of all orders. If $f\in C^\infty
(\bR),a\in \bR $ we define the Taylor series $S_f (a,x)$ for $f$ at $a$ by 
$S_f (a,x)=\sum _{n\geq 0}f^{(n)}(a)/n!(x-a)^n$.

1.Prove that for any $a,x\in \bR$ the Taylor series series  $S_f (a,x)$
is convergent to $f(x)$ for $f(x)=\exp (x)$.

2.Let $f$ be a function on $\bR$ such that $f(x)=0$ if $x\leq 0$ and
$f(x)=\exp (-1/x)$ for $x>0$.

a)Show that $f\in C^\infty (\bR)$.

b)Find the Taylor series  $S_f (0,x)$.

c)Explain why does not  $S_f (0,x)$ converge to $f(x)$.

3. Let $V$ be a finite-dimensional vector space over $\bR ,f\in C^\infty
(V),v\in V$. 

a)Define the linear  Taylor polynomial $P_1 (v,x),x\in V$ for $f$.

b))Define the quadratic  Taylor polynomial $P_2 (v,x),x\in V$ for $f$.

c))Define the   Taylor polynomial $P_n (v,x),x\in V$ for $f$.

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